where
\(\big |\widetilde{D}_{\varepsilon }\big |^{-2}:=\big (\widetilde{D}^2+\varepsilon \big )^{-1}=\big (x_1^2+x_2^2+\varepsilon \big )^{-1}\otimes {\bf 1}_2\) , and \(\varepsilon >0\) . Formula (REF ) is proved (and used) only
indirectly in [1]} but is explicitly given in [2]}. The spectral triple
\(\big (\widetilde{\rho },\widetilde{{H}}_2,\widetilde{D}\big )\)  generates a canonical
Fredholm module with Dirac phase
\(\widetilde{F}_{\varepsilon }:=\widetilde{D}_{\varepsilon }\big |\widetilde{D}_{\varepsilon }\big |^{-1}\) .
The operator \(\widetilde{F}_{\varepsilon }\)  is the key element for the
construction of the quantized calculus of the noncommutative Brillouin zone
\(\widetilde{{C}}_B\) , and the related index theory which provides the topological
interpretation of the QHE. It is worth noting that the “discrete nature” of
\(\widetilde{F}_{\varepsilon }\) , meaning that there is a (discrete) basis which diagonalizes the
Dirac phase, is a crucial ingredient in [1]},
as well as in the algebraic theory of topological
insulators [4]}, [5]} or in the
construction of the spectral localizer [6]}, [7]}, [8]}.
